| Assertion (A): The distance between the points \[(\sin \theta ,-\cos \theta )\]is 2 units. |
| Reason (R): The distance between two points \[A({{x}_{1}},{{y}_{1}})\]and \[B({{x}_{2}},{{y}_{2}})\] is given by \[AB=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}}\] |
A) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)
B) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
C) Assertion (A) is true but reason (R) is false.
D) Assertion (A) is false but reason (R) is true.
Correct Answer: D
Solution :
| [d] Reason is clearly true. |
| Now, distance between \[A(\cos \theta ,\,\sin \theta )\]and \[B(\sin \theta ,-\cos \theta )\] is given by |
| \[AB=\sqrt{{{(\sin \theta -\cos \theta )}^{2}}+{{(-\cos \theta -\sin \theta )}^{2}}}\] |
| \[=\sqrt{{{\sin }^{2}}\theta +{{\cos }^{2}}\theta -2\sin \theta \cos \theta +{{\cos }^{2}}\theta +{{\sin }^{2}}\theta +2\sin \theta \cos \theta }\]\[=\sqrt{1+1}=\sqrt{2}\]units. |
| \[\therefore \] Assertion [a] is false but reason (R) is true. |
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